Recovering a phylogenetic tree using pairwise closure operations

A fundamental task in evolutionary biology is the amalgamation of a collection P of leaf-labelled trees into a single parent tree. A desirable feature of any such amalgamation is that the resulting tree preserves all of the relationships described by the trees in P. For unrooted trees, deciding if there is such a tree is NP-complete. However, two polynomial-time approaches that sometimes provide a solution to this problem involve the computation of the semi-dyadic closure and split closure of a set of quartets that underlies P. In this paper we show that if a leaf-labelled tree T can be recovered from the semi-dyadic closure of some set Q of quartet subtrees of T , then T can also be recovered from the split-closure of Q. Furthermore, we show that the converse of this result does not hold, and resolve a closely related question posed in Böcker et al. (2000).